Simplify The Equation: 6 2 9 = 8 6 2 \frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}} 9 6 2 ​ ​ = 6 2 ​ 8 ​

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Introduction

Simplifying equations is a crucial skill in mathematics, and it's essential to understand how to approach these types of problems. In this article, we'll focus on simplifying the equation $\frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}}$. We'll break down the steps involved in solving this equation and provide a clear explanation of each step.

Understanding the Equation

The given equation is $\frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}}$. At first glance, this equation may seem complex, but with a step-by-step approach, we can simplify it and find the solution.

Step 1: Simplify the Left-Hand Side of the Equation

To simplify the left-hand side of the equation, we can start by simplifying the fraction $\frac{6 \sqrt{2}}{9}$. We can do this by finding the greatest common divisor (GCD) of the numerator and denominator.

import math
numerator = 6 * math.sqrt(2)
denominator = 9

gcd = math.gcd(int(numerator), int(denominator))

simplified_fraction = (numerator / gcd) / (denominator / gcd)
print(simplified_fraction)

Step 2: Simplify the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we can start by simplifying the fraction $\frac{8}{6 \sqrt{2}}$. We can do this by finding the greatest common divisor (GCD) of the numerator and denominator.

import math

numerator = 8
denominator = 6 * math.sqrt(2)

gcd = math.gcd(int(numerator), int(denominator))

simplified_fraction = (numerator / gcd) / (denominator / gcd)
print(simplified_fraction)

Step 3: Equate the Simplified Fractions

Now that we have simplified both sides of the equation, we can equate the two fractions.

$\frac{2 \sqrt{2}}{3} = \frac{4}{3 \sqrt{2}}$

Step 4: Cross-Multiply

To solve for the variable, we can cross-multiply the two fractions.

$2 \sqrt{2} \cdot 3 \sqrt{2} = 4 \cdot 3$

Step 5: Simplify the Equation

Now that we have cross-multiplied the two fractions, we can simplify the equation.

$12 \cdot 2 = 12$

Step 6: Solve for the Variable

Now that we have simplified the equation, we can solve for the variable.

$24 = 12$

Conclusion

In this article, we've simplified the equation $\frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}}$ using a step-by-step approach. We've broken down the steps involved in solving this equation and provided a clear explanation of each step. By following these steps, we can simplify complex equations and find the solution.

Frequently Asked Questions

• Q: What is the greatest common divisor (GCD) of two numbers? A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.
• Q: How do I simplify a fraction? A: To simplify a fraction, you can find the GCD of the numerator and denominator and divide both numbers by the GCD.
• Q: What is cross-multiplication? A: Cross-multiplication is a technique used to solve equations by multiplying both sides of the equation by the same value.

Final Answer

The final answer is $\boxed{2}$.

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Introduction

Simplifying equations is a crucial skill in mathematics, and it's essential to understand how to approach these types of problems. In this article, we'll focus on simplifying the equation $\frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}}$. We'll break down the steps involved in solving this equation and provide a clear explanation of each step.

Understanding the Equation

The given equation is $\frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}}$. At first glance, this equation may seem complex, but with a step-by-step approach, we can simplify it and find the solution.

Step 1: Simplify the Left-Hand Side of the Equation

To simplify the left-hand side of the equation, we can start by simplifying the fraction $\frac{6 \sqrt{2}}{9}$. We can do this by finding the greatest common divisor (GCD) of the numerator and denominator.

import math

numerator = 6 * math.sqrt(2)
denominator = 9

gcd = math.gcd(int(numerator), int(denominator))

simplified_fraction = (numerator / gcd) / (denominator / gcd)
print(simplified_fraction)

Step 2: Simplify the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we can start by simplifying the fraction $\frac{8}{6 \sqrt{2}}$. We can do this by finding the greatest common divisor (GCD) of the numerator and denominator.

import math

numerator = 8
denominator = 6 * math.sqrt(2)

gcd = math.gcd(int(numerator), int(denominator))

simplified_fraction = (numerator / gcd) / (denominator / gcd)
print(simplified_fraction)

Step 3: Equate the Simplified Fractions

Now that we have simplified both sides of the equation, we can equate the two fractions.

$\frac{2 \sqrt{2}}{3} = \frac{4}{3 \sqrt{2}}$

Step 4: Cross-Multiply

To solve for the variable, we can cross-multiply the two fractions.

$2 \sqrt{2} \cdot 3 \sqrt{2} = 4 \cdot 3$

Step 5: Simplify the Equation

Now that we have cross-multiplied the two fractions, we can simplify the equation.

$12 \cdot 2 = 12$

Step 6: Solve for the Variable

Now that we have simplified the equation, we can solve for the variable.

$24 = 12$

Conclusion

In this article, we've simplified the equation $\frac{6 \sqrt{2}}{9} = \frac{8}{6 \sqrt{2}}$ using a step-by-step approach. We've broken down the steps involved in solving this equation and provided a clear explanation of each step. By following these steps, we can simplify complex equations and find the solution.

Frequently Asked Questions

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can find the GCD of the numerator and denominator and divide both numbers by the GCD.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to solve equations by multiplying both sides of the equation by the same value.

Q: How do I solve for the variable in an equation?

A: To solve for the variable in an equation, you can isolate the variable by performing inverse operations on both sides of the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the significance of the x-axis and y-axis in a graph?

A: The x-axis and y-axis are the horizontal and vertical axes of a graph, respectively. They are used to measure the coordinates of points on the graph.

Final Answer

The final answer is $\boxed{2}$.